Project ForgeEngineering Mathematics
Calculus
GATE 2014
Hard
The integral ∮_C(y dx − x dy) is evaluated along the circle x²+y²=1/4 traversed in counter-clockwise direction. The integral is equal to
- A0
- B−π/4
- −π/2
- Dπ/4
Explanation
By Green's theorem, ∮(L dx + M dy) = ∬(∂M/∂x − ∂L/∂y) dA. Here L=y, M=−x, so ∂M/∂x−∂L/∂y = −1−1 = −2. The circle has radius 1/2, so area = π(1/2)² = π/4. The integral = −2×(π/4) = −π/2.
Source: GATE ME 2014 (Session 1 / Set-1) (2014)
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